Study of Behaviors of Motion Models in High-Order Systems

This paper presents several proposed motion models of high-order physical systems in three main concepts called macro-scale, regular-scale, and nano-scale. In fact, it is very difficult to find an exact numerical solution for the high-order differential equations because all numerical methods only yield the approximate solutions. In addition, loop gain is not widely used in many negative feedback systems because it is an approximation value. To overcome the limitations of the high-order differential equations and the loop gain, the waveforms of the physical periodic motions are expressed by helix functions at time variation, and the characteristics of complex functions are used to examine the behaviors of the transmission spaces and the transmission networks in the different motion models including the Earth's motion, the simple pendulum systems, and the electronic systems. Furthermore, the force of attraction and the friction or the resistance in the different scales obey the conservation law and the superposition principle; therefore, three superposition formulas are introduced to derive the transfer functions in high-order mechatronic systems. The operating regions, the effects of the overshoot phenomena, the breaking chemical bonds, and the difference between negative and positive feedbacks in these systems are also introduced. As a result, the use of complex functions, helix waves, and superposition principle leads to a complete control theory with which many behaviors of the physical systems can be explained and predicted easily.